# Group exponent properties

Is it true that for any finite cyclic group $G$, it holds that $\operatorname{exp}(G)=|G|$?

My first thought was yes, since if $G$ is cyclic, then we know it has an element of order |G|, and since any other $g\in G$, $\operatorname{order} (g)$ divides $|G|$ (from Lagrange's theorem), we have that $\operatorname{lcm}(g,|G|)=|G|$ for all $g\in G$.

But I wasn't sure, so I thought to ask, and I would also like to ask what other properties does the group exponent hold? (For some reason there isn't too much information about group exponent on the internet... not in Wikipedia, not in Proof Wiki, and certainly not in any pdf document I could find online...)

• Yes to the first question. Not sure what sort of properties you might be interested in for the second one. Jan 24, 2014 at 14:30
• Any property regarding groups, I guess, like, for example, I know that if two groups have different exponent, then they are not isomorphic. what other information about the group does the exponent provide? Jan 24, 2014 at 14:34
• OK, what about product sum of cyclic groups? what would be the exponent of $\mathbb{Z}_{n_1}\times\mathbb{Z}_{n_2}\times...\times\mathbb{Z}_{n_r}$? Jan 24, 2014 at 14:36
• That would be the lcm of the $n_i$. For abelian groups, the exponent is the same as the largest order of an element. Jan 24, 2014 at 14:38
• You might find the bounded version of Burnside's problem interesting: "If G is a finitely generated group with exponent n, is G necessarily finite?" Jan 24, 2014 at 14:41

Yes, you are correct about the property you suggest. Furthermore, in abelian groups, the exponent is equal to the largest order among the orders of the elements (so your conclusion about finite cyclic groups follows).

• The exponent divides the order of a finite group.
• The exponent of a finite group has precisely the same prime factors as order.
• The exponent of a finite group equals the product of the exponents of its Sylow subgroups.

There's not too much else to say. Hope this helps!

• (An abelian group is cyclic iff $|G|=\exp G$ might be noteworthy.)
– Pedro
Jan 24, 2014 at 15:14
• Well, the above comment is simply intended as an addition to the list.
– Pedro
Jan 24, 2014 at 15:21
• And a good one at that! Thanks, Pedro! Jan 24, 2014 at 15:23
• Thank you @amWhy, that was very useful. Jan 24, 2014 at 16:20